Find v1 in c1v1 = c2v2

This calculator finds the unknown starting volume, v1, in the dilution equation c1v1 = c2v2. Enter the starting concentration c1, target concentration c2, and final volume v2; it returns the volume of starting solution required, in the same volume unit you choose for v2. Use consistent concentration units for c1 and c2.

Formula

The dilution relationship is c1 × v1 = c2 × v2. Rearranging for the unknown starting volume gives v1 = (c2 × v2) / c1.

• c1 is the concentration of the starting solution, in mol/L for this calculator.
• v1 is the volume of starting solution to measure, in mL.
• c2 is the desired concentration after dilution, in mol/L.
• v2 is the final total solution volume, in mL.

The concentrations must use the same unit, and v1 and v2 must use the same volume unit. Here, the calculator uses mL for both volumes, so no volume conversion is needed. If your concentrations use another matching unit, such as mM for both c1 and c2, the ratio still works; interpret the result in the selected volume unit. The dilution factor c1/c2 is a helper output: for example, 10 means the target is one-tenth of the starting concentration.

Worked Example

Using the calculator’s default values: c1 = 1 mol/L, c2 = 0.1 mol/L, and v2 = 100 mL.

1. Start with the rearranged formula: v1 = (c2 × v2) / c1.
2. Substitute the values: v1 = (0.1 mol/L × 100 mL) / 1 mol/L.
3. Cancel the matching concentration units: v1 = 10 mL.
4. Check in the original relationship: 1 mol/L × 10 mL = 0.1 mol/L × 100 mL; both sides equal 10 (mol/L)·mL.

Result: v1 = 10.00 mL. The dilution factor is c1/c2 = 1 / 0.1 = 10.00. Measure 10.00 mL of the starting solution, then add suitable solvent until the total volume reaches 100 mL; do not add 100 mL of solvent to the 10 mL aliquot.

Assumptions

• The amount of dissolved substance is conserved during dilution; no reaction, precipitation, evaporation, or loss changes it.
• The starting and target concentrations refer to the same solute and are expressed in the same concentration unit.
• v2 is the final total solution volume, not the amount of solvent added.
• The arithmetic treats the stated concentrations and volumes as exact; real measurement uncertainty may affect the result.

This calculator is for general study and lab-planning arithmetic only. For bench work, follow your lab method, the safety data sheet (SDS), and your supervisor.

Step-by-Step

1. Identify the known values. Find the concentration of the stock or starting solution (c1), the concentration you want (c2), and the final total volume (v2). The unknown is v1, the portion of stock needed.
2. Make the concentration units match. Convert both concentrations to the same unit before calculating. For example, convert 100 mM to 0.100 mol/L if c1 is entered in mol/L. Do not compare a number in mM directly with a number in mol/L without converting.
3. Choose one volume unit. Use the same unit for v1 and v2. This calculator reports v1 in mL when you enter v2 in mL. If you start from a final volume in L, convert it to mL first by multiplying by 1000, or calculate with the formula using L consistently and convert the answer afterward.
4. Rearrange the equation. Begin with c1v1 = c2v2. Divide both sides by c1 to isolate v1: v1 = c2v2/c1. Make sure c1 is the starting, usually more concentrated, solution value.
5. Enter the values and calculate. Type c1, c2, and v2 into their fields. The first output is the required starting volume v1. The dilution factor is an additional check, not a replacement for the volume result.
6. Check that the answer is physically sensible. For a simple dilution, c1 should be greater than c2 and v1 should be positive and no greater than v2. If the output exceeds v2, check the concentrations and units: the requested target may be more concentrated than the starting solution, so dilution alone cannot achieve it.
7. Apply the result using the final volume correctly. The computed v1 is the amount of starting solution to include in the final mixture. Bring the mixture to v2 total volume, rather than adding v2 of solvent. Actual measuring and mixing should follow the applicable lab method.

When to Use the Equation

Use c1v1 = c2v2 when a known solution is diluted and the same dissolved substance remains in the mixture. It is commonly used to plan a single dilution from a stock solution to a chosen final concentration and volume. It can also help solve for a different missing quantity, but this page specifically calculates v1.

The equation works because concentration multiplied by volume represents the amount of solute, provided concentration and volume units are compatible. A dilution changes the volume and concentration while keeping the amount of solute the same. The formula does not determine whether a particular solution is chemically stable, compatible, or safe to handle.

Common Mistakes

• Swapping c1 and c2: c1 is the starting concentration, while c2 is the desired final concentration. Reversing them can produce an implausible volume.
• Mixing concentration units: For instance, 1 mol/L and 100 mM look numerically different but represent the same concentration. Convert before entering values.
• Using solvent volume for v2: v2 means the total final volume after the stock and solvent are combined. It is not the volume of solvent alone.
• Mixing volume units: If v2 is entered in mL, the calculated v1 is in mL. Do not treat the output as liters.
• Rounding too early: Keep a few extra digits during planning and round only to a volume that can be measured reliably. Displayed precision does not guarantee measurement accuracy.
• Trying to make a stronger solution by dilution: If c2 is greater than c1, the equation may return v1 greater than v2. Adding solvent cannot increase concentration; choose a suitable starting solution or a different preparation method.

Tips

• Write each input with its unit before substituting it into the formula. This makes unit mismatches easier to catch.
• Use the original equation as a check: calculate c1v1 and c2v2 using consistent units. The two sides should agree, apart from rounding.
• For a target below the stock concentration, the required stock volume should be a fraction of the final volume. A target one-tenth as concentrated generally means a 10-fold dilution.
• If working with very small volumes, consider whether the required volume can be measured with appropriate accuracy. A mathematically correct answer can still be impractical to measure directly.
• Record the stock concentration, target concentration, final volume, and any unit conversions alongside the result so the calculation can be checked later.

Troubleshooting

The calculator rejects an input: Check that each entry is positive and uses the stated unit. A concentration of zero cannot be used as c1 because the formula divides by c1. If a value is written in a different concentration unit, convert it to mol/L or enter both concentrations in the same unit on a consistent basis.

The answer is larger than the final volume: Compare c1 and c2. If c2 exceeds c1, simple dilution cannot reach the target. If c2 is lower than c1 but the output still looks wrong, verify that you have not swapped the fields or entered mM as if it were mol/L.

The answer seems off by a factor of 1,000: This often comes from confusing L with mL or mol/L with mmol/L (mM). Convert units explicitly, then recalculate. The calculator’s volume output follows the unit used for v2: mL in the provided setup.

The measured result does not match the target exactly: Check the concentration of the stock, the accuracy of the volume measurement, and whether v2 was brought to its final total volume. Temperature, volume changes on mixing, or an inaccurate stock concentration can also affect an actual solution. Use an appropriate measurement or validated method to confirm the result.

Frequently Asked Questions

What does v1 mean in c1v1 = c2v2?
v1 is the volume of the starting solution needed to make the chosen final volume and concentration.

How do I solve c1v1 = c2v2 for v1?
Divide both sides by c1: v1 = (c2 × v2) / c1.

Can c1 and c2 use different units?
Not directly. Convert them to the same concentration unit first; the equation relies on their ratio being correct.

Is v2 the volume of solvent I add?
No. v2 is the final total volume of the mixture. Add solvent only until the mixture reaches that total volume.

What if c2 is higher than c1?
The requested solution is more concentrated than the starting solution. Dilution alone cannot achieve it, and the calculated v1 will exceed v2.

Does the formula account for measurement error?
No. It gives an ideal arithmetic result from the values entered. The accuracy of an actual preparation also depends on the concentration and volume measurements and the method used.

Related

• Serial Dilution Calculator: Molar Concentration After n Steps
• Dilution Formula Calculator: C1V1 = C2V2 for Molarity
• Dilution Factor Calculator: Final Molarity From Volumes

in c1 | Starting concentration c1 | mol/L | 1 | 0.000001
in c2 | Target concentration c2 | mol/L | 0.1 | 0.000001
in v2 | Final solution volume v2 | mL | 100 | 0.001
out v1 | Starting volume v1 | mL | c2 * v2 / c1 | 2
out dil | Dilution factor c1/c2 | - | c1 / c2 | 2